Regularity of Solutions to Hamilton-Jacobi Equations

We formulate an Hamilton–Jacobi partial differential equation $H(x, Du(x)) = 0$ on a n dimensional manifold M , with assumptions of uni- form convexity of H(x, ·) and regularity of H in a neighborhood of {H = 0} in T ∗ M ; we define the “min solution” u, a generalized solution, which often co- incides with the viscosity solution; the definition is suited to proving regularity results about u; in particular, we prove that the closure of the set where u is not regular is a H^(n−1) –rectifiable set.